Castelnuovo-Mumford regularity and degrees of generators of graded submodules Brodmann, M Brodmann, M (2003). Castelnuovo-Mumford regularity and degrees of generators of graded submodules. Illinois Journal of Mathematics, 47(3):749-767. Abstract We extend the regularity criterion of Bayer-Stillman for a graded ideal $\mathfrak {a}$ of a polynomial ring $K[\underline {\bf x}] := K [\underline {\bf x}_0, \dots , {\bf x}_r]$ over an Abstract. We extend the regularity criterion of Bayer-Stillman

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... a graded ideal a of a polynomial ring K[x] := K[x 0 , . . . , xr] over an infinite field K to the situation of a graded submodule M of a finitely generated graded module U over a Noetherian homogeneous ring R = ⊕ n≥0 Rn, whose base ring R 0 has infinite residue fields. If R 0 is Artinian, we construct a polynomial P ∈ Q[x], depending only on the Hilbert polynomial of U , such that reg(M ) ≤ P (max{d(M ), reg(U ) + 1}), where d(M ) is the generating degree of M . This extends the regularity bound of Bayer-Mumford for a graded ideal a ⊆ K[x] over a field K to the pair M ⊆ U .